Geodesic
Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities
Journal of the European Mathematical Society, Tome 17 (2015) no. 11, pp. 2949-2975.

Voir la notice de l'article provenant de la source EMS Press

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of –Δp​u=g(u) in a smooth bounded domain Ω⊂Rn. In particular, we obtain new Lr and W1,r bounds for the extremal solution u⋆ when the domain is strictly convex. More precisely, we prove that u⋆∈L∞(Ω) if n≤p+2 and u⋆∈Ln−p−2np​(Ω)∩W01,p​(Ω) if n>p+2.
DOI : 10.4171/jems/576
Classification : 35-XX, 00-XX
Keywords: Geometric inequalities, mean curvature of level sets, Schwarz symmetrization, p-Laplace equations, regularity of stable solutions 
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     author = {Daniele Castorina and Manel Sanch\'on},
     title = {Regularity of stable solutions of $p${-Laplace} equations through geometric {Sobolev} type inequalities},
     journal = {Journal of the European Mathematical Society},
     pages = {2949--2975},
     publisher = {mathdoc},
     volume = {17},
     number = {11},
     year = {2015},
     doi = {10.4171/jems/576},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/576/}
}
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Daniele Castorina; Manel Sanchón. Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities. Journal of the European Mathematical Society, Tome 17 (2015) no. 11, pp. 2949-2975. doi : 10.4171/jems/576. http://geodesic.mathdoc.fr/articles/10.4171/jems/576/

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